1. ONE-DIMENSIONAL ANALYSIS ON BEDEVOLUTION ACCOMPANING BANK EROSION Satoru Nakanishi Hokkaido University Graduate School Kazuyoshi Hasegawa Hokkaido University Graduate School Keigo Takahashi Hokkaido University Graduate School ONE-DIMENSIONAL ANALYSIS ON BEDEVOLUTION ACCOMPANING BANK EROSION

2. 2 Table of Contents 1.Introduction Background and purpose of this study 2.Theoretical analysis Development of new model and derivation of diffusion equation 3.Movable-Bed Experiment Outline experiments and results 4.Discussion & Summary Comparison between the theory and the experiments

3. 3 INTRODUCTI ON

4. 4 Introduction Background It is necessary to understand how the river environment is affected by ARTIFICIAL POWER for long term and over the whole of a river. The relationship between river bank erosion and longitudinal profiles is an important issue in conducting river environment. There are few researches on the interaction between riverbank erosion and riverbed evolution, even though this is an important issue in river engineering and geomorphology. Purpose of This Research This research investigates the influence of riverbank erosion on the longitudinal profile. Theoretical analysis => Development of new model and derivation of diffusion equation Experimental analysis => Movable bed experiments

5. 5 Theoretical analysis

6. 6 Theoretical Model (Definition of Coordinates) A river channel pattern diagram x ： Down stream direction y ： Lateral direction from right bank z ： Upward direction L ： Channel length z b ， H m ： Bed level and bank height from base level, respectably H ： Bank height from water surface B ： River width h ： depth Simplified lateral cross-sectional model

7. 7 Theoretical Equation - Deriving Diffusion Equation Tow Dimensional Continuity Equation of Sediment Lateral sediment discharge from bank erosion is the volume of the area indicated by oblique lines. And so …. integrating the continuity equation with respect to y

8. 8 Taking MPM formula to sediment discharge Bank Erosion Rate Lateral sediment discharge the diffusion equation is derived. This equation has a differential term of the first order with respect to the longitudinal distance. Other assumptions Non-Dimensional shear stress Theoretical Equation - Deriving Diffusion Equation

9. 9 A Theoretical Solution of Diffusion Equation

10. 10 Movable-Bed Experiments

11. 11 Outline of Experiments Experimental Channel channel length L is 11 m Sediment material used as bed and bank material is uniform in diameter (0.5 mm). The cross-sectional profile is trapezoidal, and the initial angle of riverbank (in cross-section) is set as 45 O. Discharge (l/sec) Width (cm) Bank Height (cm) Slope Case1 6.020 10.51/200 Case2 10.51/500 Case3 12.51/200 Hydraulic Condition

12. 12 Photo of Experiments The Result after 1 hour into the experiment （ Case1 ） Collapse Cracks on riverbank

13. 13 Case1Case2 Case3 The Results of Experiments (Variation of Longitudinal Profile) Discharge (l/sec) Width (cm) Bank Height (cm) Slope Case1 6.020 10.51/200 Case210.51/500 Case312.51/200

14. 14 Case1Case2 Case3 The Results of Experiments (Variation of Cross-Sectional Profile)

15. 15 Discussion & Summary

16. 16 Comparison between theoretical value and experimental one (Bank Erosion Rate) Comparison between theoretical value and experimental one (Bank Erosion Rate) Experimental Value Theoretical Value

17. 17 Comparison between theoretical value and experimental one (Longitudinal Profiles) Comparison between theoretical value and experimental one (Longitudinal Profiles) METHOD K 1 was provided as constant value that was 0.03(m 2 /s). K 2 was varied in order to find the profiles that corresponded most closely to the experimental ones. Experimental values that applied the exponential function were provided as boundary condition at the upper and lower edges.

18. 18 Case1Case2 Case3 Comparison between theoretical value and experimental one (Longitudinal Profiles) Comparison between theoretical value and experimental one (Longitudinal Profiles) The theoretical profiles in each case were convex. The shapes were maintained as they moved downstream, which agrees with phenomenon observed on actual rivers as sediment mass movement.

19. 19 About K 2 Experimental Value K2 corresponded experimental profile Experimental Value (m/s) Theoretical Value (m/s) Theoretical Value K2 obtained by following Equation Although it is able to find K2 that corresponded experimental profile, it is need to discuss whether K2 corresponded to experimental value.

20. 20 A ratio of bed erosion rate to bank erosion rate Diffusion equation Bank erosion rate 3 parameters for process of long-term river change were estimated by the ratio. Discussion about erosion rate

21. 21 Discussion about sediment transport Coefficient K 2 in the diffusion equation is the coefficient of the advection term, and K 2 expresses a velocity that propagates in the downstream direction. That coefficient is dominated by parameter h/B, which means that h/B is an important parameter of riverbed evolution accompanying riverbank erosion.

22. 22 Summary 1.A diffusion equation that has a first-order differential term with respect to x has been derived 2. Experimental values have agreed closely with theoretical values, although the latter have contained slight error. 3.It has been possible to theoretically reproduce the longitudinal profiles that were obtained experimentally by the diffusion equation. The theoretical longitudinal profiles maintain their shape as they progress downstream. This phenomenon has resembled the sediment mass movement in an actual river. 4. 3 parameters for process of long-term river change have been estimated by the ratio. 5.Coefficient K 2 is dominated by parameter h/B, which means that h/B is an important parameter of riverbed evolution accompanying riverbank erosion.

23. 23 THAT ’ S ALL THANK YOU !!

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25. 25 Geomorphologic Discussion When  is greater than 1 V-shaped valley is developed When  is less than 1 U-shaped valley is developed It is possible to estimate roughly the process of geological formation by calculating the ratio.